Yes, but the level of severity is different. I was a NYS EMT, a call for SOB gets you lights and sirens, a call for not breathing gets you balls to the walls lights and sirens.
A person with SOB can be like that for a vast array of reasons- asthma, hyperventilation, anxiety, general illness, CHF, etc. But not breathing or can't breath implies that the person is literally dying at that very moment because one's heart will not continue to beat for much longer if there is no breathing. The severity is different.
SOB gets you oxygen, IV, neb or other meds, not breathing gets you a tube down your throat and more than likely CPR.